Theorem nnexpcl

Description: Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005)

	Ref	Expression
Assertion	nnexpcl	$$⊢\left(A\in \mathbb{N}\wedge N\in {\mathbb{N}}_0\right)\to A^N\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		nnsscn	$$⊢\mathbb{N}\subseteq ℂ$$
2		nnmulcl	$$⊢\left(x\in \mathbb{N}\wedge y\in \mathbb{N}\right)\to xy\in \mathbb{N}$$
3		1nn	$$⊢1\in \mathbb{N}$$
4	1 2 3	expcllem	$$⊢\left(A\in \mathbb{N}\wedge N\in {\mathbb{N}}_0\right)\to A^N\in \mathbb{N}$$